Boundary-Layer Flow Near the Trailing Edge of a Flat Plate

Source abstract

Goldstein [1] has given a solution to the boundary-layer equations for the flow just downstream from the trailing edge of a flat plate. It is shown here that this approximation is no longer valid if the nondimensional distance $\bar x/L$ from the edge is $O( R^{ - 3/8} )$. A second order correction to Goldstein’s stream function is obtained to take into account the pressure gradient induced locally in the external flow, and is found to overtake the first order term when $\bar x/L = O( R^{ - 3/ 8} )$. For an alternative derivation, a limit of the Navier-Stokes equations is taken for $R \to \infty$ with $x^ * = R^\alpha \bar x/ L$ fixed, and it is concluded that the choice $\alpha = 3/8$ corresponds to a distinguished limit in the sense that it leads to a special set of approximate equations. A solution analogous to Goldstein’s, including a second order term, is constructed for the boundary layer just upstream, from the trailing edge. The approximate equations derived in the limit for $R \to \infty$ with $x^ *$ fixed include the boundary-layer equations with pressure gradient in a sublayer of thickness $\bar y/L = O(R^{ - 5/8} )$; the conditions that the pressure gradient and flow deflection are independent of $\bar y$ for $\bar y/L = O( R^{ - 1/2} )$; and the linear-theory integral relation between pressure and flow deflection for $R^{1/2} \bar y/L \to \infty$. An iteration scheme is proposed which requires an assumed form for the derivation of the displacement thickness, with an integral method used for solution of the sublayer equations. Approximate numerical results are presented, and the possibility of obtaining higher order terms is discussed.